By Kenneth A Bowen

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Extra info for Model Theory for Modal Logic: Kripke Models for Modal Predicate Calculi

Example text

11. To see that the m constructed is strong, suppose that k, k' EK and k =1= k'. Then m(k) =1= m(k') iff {iEI: m(k) (i) =F m(k)(i)}EF. Let Z={(f,g)EI:k,k'Edom(f)}. From the argument that N has the finite intersection property, we see that Z E F. But if i = <1, g) EZ, then ULTRAPRODUCTS m(k) (i) = f (k) -+ f (k') 53 = m(k') (i) , and so {iEI: m(k)(i) -+ m(k')(i)}EF. Thus m is 1 - 1. To see that m can be made faithful when E(~) = 1), define J n as above, but require f to be faithful instead of 1 - 1.

Assurne these have been defined for n < m and let MLm - l be the language obtained by adding all these special constants to ML as new individual constants. Let 3xA be any closed formula of ML m _ 1 such that if m > 0, then A contains at least one special constant of level m - 1. 8) is a special constant oflevel m wrt Sand T; it is called the special constant for the formula 3xA. 8). Let ML2 T be the language obtained from ML by adding the special constants of level n wrt Sand T as new individual constants for all n > O.

0* is a term of sort 1. iff is an n-ary function symbol ofML(T), if ais a term ofsort 1 and b 1 , ... , bn are terms of sort 2, then f*ab 1 ... bn is a term of sort 2. if a and bare both terms of sort i, then = ab is an atomic formula, i = 1,2. if a, c are terms of sort 1 and b is a term of sort 2, then the following are atomic formulas: N*a, aR*c, and aB*b. if p is an n-ary predicate symbol of ML(T), if a is a term of sort 1 and b 1 , ... , bn are terms of sort 2, then p*ab 1 ... bn is an atomic formula.